TLDR: "rm -rf ~/.var/app/com.spotify.Client/" your Spotify home dir and start over.
I just installed the latest Flatpak (from flathub) of the Spotify client (188.8.131.526.g416cacf1) on CentOS 7. After doing this and trying to start Spotify back up, it proceeded to just open the Spotify window very quickly, and then close itself. Trying to start it up from the command line gave some more output to try to diagnose this issue.
# start the spotify client from the command line flatpak run com.spotify.Client /app/extra/bin/spotify: /app/lib/libcurl-gnutls.so.4: no version information available (required by /app/extra/bin/spotify) [spotifywm] attached to spotify [spotifywm] spotify window found
After doing this, the window was nowhere to be found. I thought it might be and issue with the file it could not find which was the "libcurl-gnutls" error above. I eventually found out this had nothing to do with the issue and it could be ignored. Eventually the only thing that fixed this issue was removing my whole Spotify home dir (which wipes out any downloaded files, settings, and your login info). Removing the dir is easy as just deleting directory "rm -rf ~/.var/app/com.spotify.Client/". Then run the command to start up the Flatpak again "flatpak run com.spotify.Client". It should start right up and you will have to login again.